I have to admit to becoming slightly obsessed with perhaps one of the strangest mammals ever to have walked on the planet. I encountered them on a shoot with the BBC's natural history unit earlier this year, making a series about giant ice-age beasts. Turning off a long straight highway through the Safford basin in eastern Arizona, I met palaeontologist Dave Gillette. As the sun sank low, we cut into the desert, off-road, kicking up plumes of dust from our tyres. Eventually, we came to a halt, and in a pit with an A-frame standing over it, I could see a huge lump of white plaster. Except, of course, this was not just any lump of plaster – it contained a fossilised glyptodont. It was huge.

The next morning I headed back to the site at sunrise; the snow on top of the western mountains a rosy pink. It was around 6.30 when I arrived, and the desert air was distinctly chilly. The palaeontologists were already hard at work with no time to spare: the plaster bandages set rock-hard very quickly. By 10.30am, it was already getting uncomfortably hot.

Two million years ago, when the creature in the plaster jacket had been alive, the landscape here would have been completely different: instead of a desert, it would have been quite lush and watery. Fossil glyptodonts are often found alongside the remains of those overgrown, swamp-living rodents, capybaras.

Dave had started working on glyptodonts about 40 years ago, for his postgraduate thesis. After a long hiatus, when he'd often found himself working on dinosaurs, he was getting reacquainted with his old friends. His description made them sound like fictional, fantastical beasts: the largest were the size of a VW Beetle, weighing in at 800kg. They were covered in a bony shell, made of hexagonal plates called scutes or osteoderms, which developed in the skin of the animal. But they also had bristles that penetrated through the armour, probably for sensation.

"It's so strange – you expect a mammal to be hairy," I ventured.

"Well, they would have had furry necks and bellies," replied Dave.

They needed to be well-armoured: roaming the area at the same time were some formidable predators. Despite their armour, it seemed at least one had fallen prey to a scimitar cat: there was a famous discovery of a glyptodont skull bearing two elliptical holes where a pair of long teeth had pierced it, Dave told me.

The following day, the half-excavated and plaster-jacketed glyptodont from the first site was lifted on to the back of a truck, and taken to the Arizona Museum of Natural History in Mesa. There, Dave showed me a cast of a glyptodont skull: the strangest mammal skull I think I've ever seen. It looked like a work of science fiction.

Glyptodonts come from an ancient family, appearing about 40 million years ago and disappearing off the face of the earth 11,000 years ago. They evolved in South America, reaching Mexico about 4 million years ago, and then spreading into what is now the southern United States. The characteristic feature of this long-lived family of animals was their carapace (upper dorsal shell), made of scutes. For glyptodont aficionados such as Dave, the particular rosette patterns in the scutes could help with identifying different species of glyptodonts.

There was an impressive collection of scutes in the museum lab. They looked like thick, bony wafers, and it seemed so odd that they were from a mammal. But we have a few bones in our bodies that form, under the skin, in a similar way: the flat bones forming the vault of our skulls, and even our clavicles. Most mammals stick with making their skulls and their clavicles (if they have them) in this way, but glyptodonts have done this weird thing and made themselves a whole body-case out of this dermal bone. In fact, they're not the only animals to have done this, but they're among a select few, including extinct ground sloths and modern armadillos.

As well as developing in the same way as the human skull, osteoderms have a similar structure, with a layer of spongy bone sandwiched between more compact layers. (Turtle shells are also similar – and, in Dave's lab, I was startled to see just how much fragments of fossil turtle shell looked like bits of human skull. I'm not surprised that many archaeologists are taken in, mistaking pieces of turtle shell for human skull, much to the ill-disguised amusement of the palaeontologists.)

So why are the glyptodonts no longer with us? As with many giant beasts from the ice age, it seems more than a coincidence that their disappearance follows the first appearance of humans in a continent. But it's still a massive assumption to blame the demise of all these animals – mammoths, mastodons, sabre-toothed cats, American lions, giant ground sloths, glyptodonts – on our ancestors. The climate was changing dramatically – couldn't that have been to blame? As usual, if we're looking for simple answers, we're quickly drawn into nuance and complexity. Recent genetic analyses suggest that humans may have been more culpable in some cases, climate in others.

Whatever the reason for the loss of all these animals (and, who knows, perhaps elucidating it might help us to better protect endangered species today), the golden age of mammals is over. There are still some behemoths in Africa, but the magnificent ice-age megafauna are preserved only as fossils in Asia, Australia and the Americas. It is a shame. It was incredible to meet the fossilised remains of a glyptodont, but I would love to have met one of those most bizarre of creatures in the flesh.